Predictive coding (PC) is an energy-based learning algorithm that performs iterative inference over network activities before updating weights. Recent work suggests that PC can converge in fewer learning steps than backpropagation thanks to its inference procedure. However, these advantages are not always observed, and the impact of PC inference on learning is not theoretically well understood. To address this gap, we study the geometry of the PC weight landscape at the inference equilibrium of the network activities. For deep linear networks, we first show that the equilibrated PC energy is equal to a rescaled mean squared error loss with a weight-dependent rescaling. We then prove that many highly degenerate (non-strict) saddles of the loss including the origin become much easier to escape (strict) in the equilibrated energy. Experiments on both linear and non-linear networks strongly validate our theory and further suggest that all the saddles of the equilibrated energy are strict. Overall, this work shows that PC inference makes the loss landscape of feedforward networks more benign and robust to vanishing gradients, while also highlighting the fundamental challenge of scaling PC to very deep models.

Attention mechanisms play a crucial role in cognitive systems by allowing them to flexibly allocate cognitive resources. Transformers, in particular, have become a dominant architecture in machine learning, with attention as their central innovation. However, the underlying intuition and formalism of attention in Transformers is based on ideas of keys and queries in database management systems. In this work, we pursue a structural inference perspective, building upon, and bringing together, previous theoretical descriptions of attention such as; Gaussian Mixture Models, alignment mechanisms and Hopfield Networks. Specifically, we demonstrate that attention can be viewed as inference over an implicitly defined set of possible adjacency structures in a graphical model, revealing the generality of such a mechanism. This perspective unifies different attentional architectures in machine learning and suggests potential modifications and generalizations of attention. Here we investigate two and demonstrate their behaviour on explanatory toy problems: (a) extending the value function to incorporate more nodes of a graphical model yielding a mechanism with a bias toward attending multiple tokens; (b) introducing a geometric prior (with conjugate hyper-prior) over the adjacency structures producing a mechanism which dynamically scales the context window depending on input. Moreover, by describing a link between structural inference and precision-regulation in Predictive Coding Networks, we discuss how this framework can bridge the gap between attentional mechanisms in machine learning and Bayesian conceptions of attention in Neuroscience. We hope by providing a new lens on attention architectures our work can guide the development of new and improved attentional mechanisms.